Open Questions For The Smarandache Function

Open Questions For The Smarandache Function Mihaly Bencze Brasov, Romania Let Sen) be the Smarandache function. I propose the following open questions: 1) Solve the following equation in integers:

2) Solve the following equation in integers:

3) Solve the following equation in integers:

S(d(n) + cr(n» = d(S(n» + cr(S(n».

4) Solve the following equation in integers:

S(a*d(n) + b*cr(n) + c*€j>(n) + d*\jJ(n»

=

a*d(S(n» + b*cr(S(n» + c*cp(S(n» + d*\jJ(S(n».

5) Solve the following equation in integers: n

n

II S(k)*~(n) k=l

6) Solve the following equation in integers:

± S{l) ± S(2) ± ... ± Sen) = H(n(n+ 1»/2).

7) Solve the following equation in integers:

8) Solve the following equation in integers:

201

S( ± ~(1) ± ~(2) ± ... ± ~n»

= S(~«n(n+ 1»/2).

9) Solve the following equation in integers: S(± d(1) ± d(2) ± ... ± den) ) = S( d«n(n+ 1»/2).

10) Solve the following equation in integers: S(± 0'(1) ± 0'(2) ± ... ± O'(n» = S(cr(n(n+ 1»)/2).

11) Solve the following equation in integers:

1

1

+ S(1)

+

n

+

S(2)

Sen)

S«n(n+ 1»/2)

12) Solve the following equation in integers: S(1 *2) + S(2*3) + ... + S(n(n+ 1»

=

S«n(n+ 1)(n+2»/3).

13. Let uk(n) be the first k digits ofn and ~p(n) the last p digits ofn. Determine all integer 5-tuples (n,m,r,k,p) for which:

and

14) Determine all integer 5-tuples (n,m,r,k,p) for which:

and

15) Determine all integer pairs (n,k) for which:

202

16) Detennine all integer pairs (o,p) for which:

17) Find all integer pairs (o,k) such that

18) Find all integer pairs (n,p) such that

19) Let Pn be the n-th prime number. Detennine all integer triples (n,k,p) for which

and

20) Find all integer pairs (a,b) such that

a*S(b) + h*S(a)

a+b

21) Solve the following in integers: ±S(cr(l»±S(cr(2»± .. ±S(cr(n» = ±cr(±S(1)±S(2)± ... ±S(n».

22) Solve the following in integers: Uk(n) = Sen)

and

~p(n) =

Sen).

23) Solve the following in integers: Uk(n!) = SCm) and ~Vn!) = SCm).

203